Physical Solutions of The Symmetric Teleparallel Gravity in Two-Dimensions

Physical Solutions of The Symmetric Teleparallel Gravity in Two-Dimensions

Muzaffer Adak

Department of Physics, Faculty of Arts and Sciences, Pamukkale University, 20100 Denizli, Turkey

madak@pau.edu.tr

Tekin Dereli

Department of Physics, Ko¸c University 34450 Sarıyer-Istanbul, Turkey

tdereli@ku.edu.tr

Abstract

A 2D symmetric teleparallel gravity model is given by a generic 4- parameter action that is quadratic in the non-metricity tensor. Vari- ational field equations are derived. For a particular choice of the cou- pling parameters and in a natural gauge, we give a Schwarzschild type solution and an inflationary cosmological solution.

1 Introduction

In order to investigate the physical implications of General Relativity (GR) some simple limiting cases are often sought. The low energy, static limit is attained by assuming the existence of a time-like Killing vector. However, such configurations have no dynamics. Symmetric limits at arbitrary energy scales on the other hand are obtained by assuming one or more space-like Killing vectors. For example, the assumption of spherical symmetry reduces the gravitational action upon integrating over the angular variables to an effective 2D gravity model in (t, r) coordinates. Such 2D gravity models have recently been used much to discuss black hole dynamics, quantized gravity or numerical relativity [1]. It is well known that Einsteinean gravity in 2D has no dynamical degrees of freedom. One way of making this theory dynamical is to couple a dilaton scalar. It is also possible to introduce further dynamical degrees of freedom by going over to a non-Riemannian space-time by the inclusion of torsion and non-metricity. Generalized 2D gravity models with curvature and torsion and teleparallel theories with only torsion have been studied a lot [2, 3]. On the other hand 2D models including non-metricity received less attention [4, 5]. The symmetric teleparallel gravity (STPG) with zero-curvature and zero-torsion has been introduced relatively recently [6, 7, 8]. Here we consider 2D symmetric teleparallel gravity models with the most general 4-parameter action that is quadratic in the non-metricity tensor [9]. For a particular choice of parameters we have shown that the corresponding set of field equations admit both the Schwarzschild solution and an inflationary cosmological solution. It is remarkable that the STPG equivalent of Einstein gravity in 2D does not admit these solutions.

2 Mathematical Preliminaries

The triple {M, g,∇} denotes the space-time where M is a 2-dimensional differentiable manifold, g is a non-degenerate Lorentzian metric and∇is a linear connection. g can be written in terms of the co-frame 1-forms

g=g_αβdx^α⊗dx^β =η_abe^a⊗e^b (1) where {e^a} is an orthonormal co-frame and {dx^α} are the co-ordinate co- frame 1−forms. We take η_ab = Diag(−,+) and g_αβ are the co-ordinate components of the metric. The dual orthonormal frame is determined from the relations e^b(X_a) = ı_ae^b = δ^b_a. Similarly, dx^β(∂_α) = ı_αdx^β = δ_α^β. Here ı denotes the interior product operator that maps anyp-form to a (p−1)- form. We set space-time orientation by the choice²₀₁= +1 or ∗1 =e⁰∧e¹

where ∗ is the Hodge star. Finally, the connection is specified by a set of connection 1-forms{Λ^a_b}. The non-metricity 1-forms, torsion 2-forms and curvature 2-forms are defined through the Cartan structure equations:

Q_ab = −1

2Dη_ab= 1

2(Λ_ab+ Λ_ba), (2) T^a = De^a=de^a+ Λ^a_b∧e^b , (3) R^a_b = DΛ^a_b =dΛ^a_b+ Λ^a_c∧Λ^c_b (4) Bianchi identities follow as their integrability conditions:

DQ_ab = 1

2(R_ab+R_ba), (5)

DT^a = R^a_b∧e^b, (6)

DR^a_b = 0. (7)

If the non-metricity is non-vanishing, special attention is due in lowering and raising indices under covariant exterior derivative. We make use of the identities

D∗e_a = −Q∧ ∗e_a+²_abT^b (8)

D∗e_ab = −²_abQ (9)

whereQ= Λ^a_a=Q^a_ais called the Weyl 1-form. The full connection 1-forms can be decomposed uniquely as follows [4, 5]:

Λ^a_b =ω^a_b+K^a_b+q^a_b+Q^a_b (10) whereω^a_b are the Levi-Civita connection 1-forms satisfying

ω^a_b∧e^b =−de^a, (11) K^a_b are the contortion 1-forms satisfying

K^a_b∧e^b =T^a, (12)

and q^a_b are the anti-symmetric tensor 1-forms defined by

q_ab=−(ı_aQ_bc)∧e^c+ (ı_bQ_ac)∧e^c. (13) In this decomposition the symmetric part

Λ_(ab) =Q_ab (14)

while the anti-symmetric part

Λ_[ab]=ω_ab+K_ab+q_ab. (15) In writing down gravity models it easily becomes complicated to keep all the components of Q^a_b. Therefore, people sometimes deal only with certain irreducible parts of that. To obtain the irreducible decomposition of non-metricity tensor under the 2D Lorentz group, firstly we write

Q_ab= Q_ab

|{z}

trace−f ree part

+ 1 2η_abQ

| {z }

trace part

. (16)

Then

Q_ab=⁽¹⁾Q_ab+⁽²⁾Q_ab+⁽³⁾Q_ab (17) in terms of

(2)Q_ab = 1

2[(ı_aΛ)e_b+ (ı_bΛ)e_a−η_abΛ], (18)

(3)Q_ab = 1

2η_abQ , (19)

(1)Q_ab = Q_ab−⁽²⁾Q_ab−⁽³⁾Q_ab (20)

where Λ := (ı_bQ^b_a)e^a.These irreducible components satisfy

η_ab⁽¹⁾Q^ab =η_ab⁽²⁾Q^ab= 0, ı_a⁽¹⁾Q^ab= 0, e_a∧⁽¹⁾Q^ab = 0. (21) Thus they are orthogonal in the following sense:

(i)Q^ab∧ ∗^(j)Q_ab =δ^ijN_ij (no summation overij) (22) whereδ^ij is the Kronecker symbol andN_ij are certain 2-forms. We calculate

(1)Q^ab∧ ∗⁽¹⁾Q_ab = Q^ab∧ ∗Q_ab−⁽²⁾Q^ab∧ ∗⁽²⁾Q_ab−⁽³⁾Q^ab∧ ∗⁽³⁾Q_ab ,

(2)Q^ab∧ ∗⁽²⁾Q_ab = (ı^aQ_ac)(ı_bQ^bc)∗1 +1

4Q∧ ∗Q−(ı_aQ)(ı_bQ^ab)∗1,

(3)Q^ab∧ ∗⁽³⁾Q_ab = 1

2Q∧ ∗Q . (23)

3 Symmetric Teleparallel Gravity

We formulate STPG by a variational principle from an action I =

Z

M

³

L+R^a_bρ_a^b+T^aλ_a

´

(24) whereLis a Lagrangian density 2-form quadratic in the non-metricity ten- sor,ρ_a^bandλ_aare the Lagrange multiplier 0-forms imposing the constraints

R^a_b = 0 , T^a= 0. (25)

The gravitational field equations are derived from (24) by independent vari- ations with respect to the connection {Λ^a_b} and the orthonormal co-frame {e^a} (and the Lagrange multipliers):

λ_a∧e^b+Dρ_a^b =−Σ_a^b , (26)

Dλ_a=−τ_a (27)

where Σ_a^b = _∂Λ^∂La

b andτ_a= _∂e^∂La. In principle (26) is solved for the Lagrange multipliersλ_a andρ_a^b and substituted in (27) that governs the dynamics of the gravitational fields. It is important to notice thatDλ_a rather than the Lagrange multipliers themselves appear in (27). Thus we must be calculating Dλ_a directly and that we can do by taking the covariant exterior derivative of (26):

Dλ_a∧e^b=−DΣ_a^b. (28) We used above the constraints

T^a= 0 , D²ρ_a^b=R^b_c∧ρ_a^c−R^c_a∧ρ_c^b = 0 (29) where the covariant exterior derivative of a (1,1)-type tensor is given by

Dρ_a^b=dρ_a^b+ Λ^b_c∧ρ_a^c−Λ^c_a∧ρ_c^b . (30) Finally we arrive at the gravitational field equations

DΣ_a^b−τ_a∧e^b = 0. (31) We now write down the most general Lagrangian density 2-form which is quadratic in the non-metricity tensor [9]:

L= 1 2κ

" ₃ X

I=1

k_I^(I)Q_ab∧ ∗^(I)Q^ab+k₄

³(2)Q_ab∧e^b

´

∧ ∗

³(3)Q^ac∧e_c

´# . (32)

where k₁, k₂, k₃, k₄ are dimensionless coupling constants and we also intro- duced κ = ^8πG_c3 with G being Newton’s gravitational constant. Inserting (23) into (32) we find

L= 1 2κ

h

c₁Q_ab∧ ∗Q^ab+c₂(ı_aQ^ac)(ı^bQ_bc)∗1 +c₃Q∧ ∗Q+c₄(ı_aQ)(ı_bQ^ab)∗1 i

(33) where we defined new coupling constants:

c₁ = k₁,

c₂ = −k₁+k₂ , c₃ = −3

4k₁+1 4k₂+1

2k₃+1 4k₄ , c₄ = k₁−k₂−1

2k₄. (34)

We remark at this point that STP equivalent of the Einstein-Hilbert Lagrangian density will be obtained for the choicec₁ =−1, c₂ = 0, c₃ =

−1, c₄= 2. We wish to show this briefly. Firstly we use the decomposition of the full connection (10), withK^a_b = 0,

Λ^a_b=ω^a_b+ Ω^a_b where Ω^a_b =Q^a_b+q^a_b . (35) By substituting that intoR^a_b(Λ) we decompose the non-Riemannian curva- ture as the follows:

R^a_b(Λ) = dΛ^a_b+ Λ^a_c∧Λ^c_b

= R^a_b(ω) +D(ω)Ω^a_b+ Ω^a_c∧Ω^c_b. (36) Here R^a_b(ω) is the Riemannian curvature 2-form and D(ω) is the covari- ant exterior derivative with respect to the Levi-Civita connection. To set R^a_b(Λ) = 0 for STP space-time yields the Einstein-Hilbert Lagrangian 2- form

L_EH = R^a_b(ω)∧ ∗e_a^b

= −[D(ω)Ω^a_b]∧ ∗e_a^b−Ω^a_c∧Ω^c_b∧ ∗e_a^b (37) Here after using the equality

d(Ω^a_b∧ ∗e_a^b) = [D(ω)Ω^a_b]∧ ∗e_a^b−Ω^a_b∧[D(ω)∗e_a^b] (38)

we discard the exact form and noticeD(ω)∗e_a^b = 0 because T^a(ω) = 0 and Q^a_b(ω) = 0 (see eq.(9)). Thus, up to a closed form

L_EH = 1

2κΩ^a_c∧Ω^c_b∧ ∗e_a^b

= 1

2κ(Q^ac+q^ac)∧(Q_cb+q_cb)∧ ∗e_a^b

= 1

2κ(Q^ac∧Q_cb+q^ac∧q_cb)∧ ∗e_a^b

= 1

2κ[−Q_ab∧ ∗Q^ab−Q∧ ∗Q+ 2(ı_bQ)(ı_aQ^ab)∗1] (39) whereκ is gravitational coupling constant.

4 Solutions

We obtain the following contributions to the variational field equations (31) coming from (33):

Σ_a^b = X4

i=0

c_iⁱΣ_a^b , τ_a= X4

i=0

c_i ⁱτ_a (40) where

1Σ_ab = 2∗Q_ab , (41)

2Σ_ab = ı^cQ_ac∗e_b+ı^cQ_bc∗e_a , (42)

3Σ_ab = 2η_ab∗Q , (43)

4Σ_ab = 1

2(ı_aQ)∗e_b+1

2(ı_bQ)∗e_a+η_ab(ı_cQ^cd)∗e_d, (44)

1τ_a = −(ı_aQ^bc)∧ ∗Q_bc−Q^bc∧(ı_a∗Q_bc), (45)

2τ_a = (ı_bQ^bc)(ı^dQ_cd)∗e_a−2(ı_aQ^bd)(ı^cQ_cd)∗e_b , (46)

3τ_a = −(ı_aQ)∧ ∗Q−Q∧(ı_a∗Q), (47)

4τ_a = (ı_bQ)(ı_cQ^bc)∗e_a−(ı_aQ)(ı_cQ^bc)∗e_b−(ı_bQ)(ı_aQ^bc)∗e_c.(48) One can consult Ref.[8] for the details of variations.

A generic class of solutions will be obtained in the coordinate frame e^α = dx^α in which Λ^α_β = 0. We call this the natural or inertial gauge choice:

R^α_β =dΛ^α_β+ Λ^α_γ∧Λ^γ_β = 0, (49) T^α =d(dx^α) + Λ^α_β∧dx^β = 0, (50) Q_αβ =−1

2Dg_αβ =−1

2dg_αβ 6= 0. (51)

After a frame transformation via the zweibein e^a = h^a_αdx^α and setting Λ^a_b=h^a_αΛ^α_βh^β_b+h^a_αdh^α_b we obtain the orthonormal components

R^a_b =h^a_αR^α_βh^β_b = 0, T^a=h^a_αT^α= 0, Q_ab=Q_αβh^α_ah^β_b 6= 0. (52) This shows that in the natural gauge, the field equations may be solved just by a metric ansatz.

4.1 Static solutions First we consider

g=−f²(r)dt²+g²(r)dr². (53) The co-ordinate components of the metric and the zweibein reads, respec- tively, as follows

g_αβ =

µ −f² 0 0 g²

¶

, h^a_α=

µ f 0 0 g

¶

. (54)

The coordinate components of non-metricity is found to be Q_αβ =−1

2dg_αβ = Ã f f⁰

g e¹ 0 0 −g⁰e¹

!

(55) where prime denotes the derivative with respect to r. The orthonormal components are obtained through a frame transformation

Q_ab =Q_αβh^α_ah^β_b= Ã f⁰

f ge¹ 0 0 −_g^g2⁰e¹

!

. (56)

In the above configuration the only non-trivial field equation comes from the trace of (31):

dΣ^a_a= 0 (57)

that reads explicitly α

"µ f⁰

f

¶₀ +

µf⁰ f

¶₂# +β

"µ g⁰

g

¶₀

− µg⁰

g

¶₂#

+ (β−α)f⁰ f

g⁰

g = 0 (58) whereα= 2c₁+ 4c₃+c₄, β = 2c₁+ 2c₂+ 4c₃+ 3c₄ . Mathematically, this equation has infinitely many solutions because there are two functions and

only one equation. That is, giveng(r) we determine the correspondingf(r).

Physically, however, due to the observational success of the Schwarzschild solution of GR, we seek solutions of the form

f(r) = µ

a+b r

¶_p

, g(r) = µ

a+ b r

¶_q

. (59)

Then (58) becomes

(p−q+ 1 + 2ar

b )(pα+qβ) = 0 (60) and therefore we let β = −^p_qα. We note here that STP Einstein-Hilbert action corresponds to the choice p = 0 with arbitrary q. Finally with p = 1/2, β = α and a suitable choice of a and b, the Schwarzschild metric is obtained:

g=−(1−2m

r )dt²+ dr²

(1− ^2m_r ). (61)

4.2 Cosmological solutions

For cosmological solutions we start with the metric

g=−dt²+R²(t)dr² (62)

whereR(t) is the expansion function. In the natural gauge, the non-metricity has just one non-zero component

Q₁₁=−R˙

Re⁰ , others = 0 (63)

where dot denotes the derivative with respect to t. For this configuration, the only non-trivial part of (31) is, again, its trace;dΣ_a^a= 0 which reads

α



 ÃR˙

R

!_. +

ÃR˙ R

!₂

= 0. (64)

This equation accepts two classes of solution, as well.

1. Forα6= 0, we obtain

R(t) =a+bt (65)

whereaandbintegration constants. The case of STP Einstein-Hilbert action belongs here.

2. Forα= 2c₁+ 4c₃+c₄ = 2k₃+ ¹₂k₄ = 0, we set ÃR˙

R

!_. +

ÃR˙ R

!₂

=S(t) (66)

where S(t) is any t dependent function. This result contains many familiar solutions. For example;

• ForS =H²: Hubble constant, we obtain the Robertson-Walker metric satisfying the perfect cosmological principle. A co-ordinate transformation tan^√₂^kr= ^√₂^kρ, takes the metric to

ds²=−dt²+e^2Ht dρ²

(1 + ¹₄kρ²)² . (67)

5 Conclusion

In this paper we investigated the symmetric teleparallel gravity in two- dimensions. After giving the orthogonal, irreducible decomposition of the non-metricity tensor under the 2D Lorentz group, we wrote down the most general four-parameter Lagrangian density 2-form quadratic in the non- metricity tensor. We obtained the variational field equations and for any particular choice of coupling constants, we have shown that the correspond- ing field equations in a natural gauge admit both the static Schwarzschild solution and an inflationary cosmological solution.

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